Value of polygamma at 1

From specialfunctionswiki
Revision as of 08:08, 11 June 2016 by Tom (talk | contribs)
Jump to: navigation, search

Theorem

The following formula holds for $m=1,2,3,\ldots$: $$\psi^{(m)}(1)=(-1)^{m+1} m! \zeta(m+1),$$ where $\psi^{(m)}$ denotes the polygamma, $m!$ denotes the factorial, and $\zeta$ denotes the Riemann zeta function.

Proof

Reference